Wrong PariGP data for Newform orbit 80.5.h.b
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The PariGP data on the page https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/80/5/h/b/
for Newform orbit 80.5.h.b, namely,
[N,k,chi] = [80,5,Mod(79,80)]
is not correct.
It is the same as the data for Newform orbit 80.5.h.a
[N,k,chi] = [80,5,Mod(79,80)]
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/80/5/h/a/
This data does appear to be correct for Newform orbit 80.5.h.a, when I use it in PariGP.
It would be great if someone could fix the PariGP data for Newform orbit 80.5.h.b, as I would like to work with it in PariGP.
The data for Magma and SageMath may be wrong also.
The data is not incorrect, in all three cases the code snippets split the space, 80.5.h into three newform subspaces, one of which is 80.5.h.b. Pari/GP, Magma, and Sage do not guarantee any particular ordering of the newform subspaces (which may even vary from one call to the next). This is why the code snippets/downloads include the comment Warning: index may be different, there is no way to know in advance what the index will be.
To find a particular newform you need to compare traces (or in this case, just look at the coefficient field which is Q(i) for 80.5.h.b).
We could automate this by adding a code that does this trace comparison for you. I have created a separate issue for this, see #6060